In a multivariate "errors in variables" regression model, the unknown mean vectors u1i: p × 1, u2i: r × 1 of the vector observations x1i, x2i, rather than the observations themselves, are assumed to follow the linear relation: u2i = α + Bu1i, i = 1,2,⋯, n. It is further assumed that the random errors ei = xi - ui, x'i = (x'1i, x'2i), u'i = (u'1i, u'2i), are i.i.d. random vectors with common covariance matrix Σe. Such a model is a generalization of the univariate (r = 1) "errors in variables" regression model which has been of interest to statisticians for over a century. In the present paper, it is shown that when Σe = σ2I p+r, a wide class of least squares approaches to estimation of the intercept vector α and slope matrix B all lead to identical estimators α̂ and B̂ of these respective parameters, and that α̂ and B̂ are also the maximum likelihood estimators (MLE's) of α and B under the assumption of normally distributed errors ei. Formulas for α̂, B̂ and also the MLE's Û1 and σ̂2 of the parameters U1 = (u11, ⋯, u1n) and σ2 are given. Under reasonable assumptions concerning the unknown sequence {u1i, i = 1,2,⋯}, α̂, B̂ and r-1(r + p)σ̂2 are shown to be strongly (with probability one) consistent estimators of α, B and σ2, respectively, as n → ∞, regardless of the common distribution of the errors ei. When this common error distribution has finite fourth moments, α̂, B̂ and r-1(r + p)σ̂2 are also shown to be asymptotically normally distributed. Finally large-sample approximate 100(1 - ν)% confidence regions for α, B and σ2 are constructed.
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© 1981 Institute of Mathematical Statistics
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